Monte Carlo Minimization Research Articles

Kinked-Helices Model of the Nicotinic Acetylcholine Receptor Ion Channel and Its Complexes with Blockers: Simulation by the Monte Carlo Minimization Method

A model of the nicotinic acetylcholine receptor ion channel was elaborated based on the data from electron microscopy, affinity labeling, cysteine scanning, mutagenesis studies, and channel blockade. A restrained Monte Carlo minimization method was used for the calculations. Five identical M2 segments (the sequence EKMTLSISVL 10LALTVFLLVI 20V) were arranged in five-helix bundles with various geometrical profiles of the pore. For each bundle, energy profiles for chlorpromazine, QX-222, pentamethonium, and other blocking drugs pulled through the pore were calculated. An optimal model obtained allows all of the blockers free access to the pore, but retards them at the rings of residues known to contribute to the corresponding binding sites. In this model, M2 helices are necessarily kinked. They come into contact with each other at the cytoplasmic end but diverge at the synaptic end, where N-termini of M1 segments may contribute to the pore. The kinks disengage α-helical H-bonds between Ala 12 and Ser 8. The uncoupled lone electron pairs of Ser 8 carbonyl oxygens protrude into the pore, forming a hydrophilic ring that may be important for the permeation of cations. A split network of H-bonds provides a flexibility to the chains Val 9-Ala 12, the numerous conformations of which form only two or three intrasegment H-bonds. The cross-ectional dimensions of the interface between the flexible chains vary essentially at the level of Leu 11. We suggest that conformational transitions in the chains Val 9-Ala 12 are responsible for the channel gating, whereas rotations of more stable α-helical parts of M2 segments may be necessary to transfer the channel in the desensitized state.

Global minima for transition metal clusters described by Sutton–Chen potentials

Using a Monte Carlo minimization approach we report the global minima for metal clusters modelled by the Sutton–Chen family of potentials containing up to 80 atoms. The resulting structures are discussed in the light of both experimental and theoretical data for clusters of the appropriate elements.

Efficiency of simulated annealing and the Monte Carlo minimization method for generating a set of low energy structures of peptides

Peptides in solution are in most cases random coils but under special solvent conditions can produce in NMR experiments medium- and long-range nuclear Overhauser effect intensities (NOEs). However sometimes the latter are only compatible with a molecule that populates several states in thermodynamic equilibrium. A new methodology for analysing such data is based on an extensive conformational search for the global energy minimum (GEM) as well as the other minima in a certain energy range above the GEM. A convenient and potentially efficient method for such a search is simulated annealing (SA) in which the energy of selected conformations is minimized during the simulation. We apply such SA procedure to ECEPP models of the pentapeptides Leu- and Met-enkephalin and find that it is significantly less efficient than the Monte Carlo minimization method of Li and Scheraga.

Docking by Monte Carlo minimization with a solvation correction: Application to an FKBP?substrate complex

A Monte Carlo docking procedure that combines random displacements of the substrate and protein side chains with minimization of the enzyme—substrate complex is described and applied to finding the binding mode of the blocked tetrapeptide N-acetyl-Leu-Pro-Phe-methylamide to the FK506 binding protein (FKBP). The tetrapeptide, an analog of the preferred FKBP substrate, and the FKBP binding site are flexible during the docking procedure. The twisted-imide transition-state form of the substrate is used during docking. The enzyme charges are scaled individually to account for solvent screening of specific binding site residues during the Monte Carlo sampling. To evaluate the relative binding free energies of the resulting structures, a rapid method for calculating polar and nonpolar solvation effects is introduced. Accurate electrostatic solute—solvent energies are calculated by solving the finite-difference linearized Poisson—Boltzmann equation; nonpolar contributions to the stability of the different conformers are estimated by the free energy of cavity formation, which is obtained from the molecular surface, and the solute—solvent van der Waals energy, which is calculated with a continuum approach. In the conformation of the enzyme—substrate complex with the lowest free energy, the tetrapeptide is bound as a type VIa proline turn with solvent accessible ends to permit longer polypeptide chains to act as substrates. Except for the imide carbonyl, which is involved in polar interactions with aromatic side chains of the FKBP binding site, all of the seven potential hydrogen bond donors or acceptors of the tetrapeptide are satisfied. The FKBP binding site has a similar conformation in the substrate complex as in the FKBP-FK506 cocrystal structure, except for the predicted reorientation of the Tyr 82 hydroxyl, which plays an important role in substrate binding. The present model for the FKBP—substrate complex is in agreement with the recently determined crystal structure of a cyclic peptide—FK506 hybrid bound to FKBP and supports the structure obtained previously by iterative model building. In addition, it is consistent with the observed effects of FKBP point mutations on the enzyme activity. The approach described here should be useful, in general, for the prediction of the structure of a molecule in solution or as part of a complex. It provides for the effective sampling of conformational space and for the inclusion of solvent effects. © 1997 by John Wiley & Sons, Inc.

Efficiency of the Local Torsional Deformations Method for Identifying the Stable Structures of Cyclic Molecules

A new method for generating the low-energy structures of a chain molecule was proposed recently by us. This is a stochastic process where at each step an energy-minimized structure is changed by carrying out several local torsional deformations (LTDs) along the chain, which temporarily disrupt neighbors of the rotated bonds. The energy is then minimized and the disrupted bonds return to their usual geometry (in terms of bond lengths and angles) while the chain assumes a new conformation. This conformation is accepted (and then deformed) or rejected with the help of a “selection procedure” that gives preference to accepting the lower energy structures and, thus, directs the search toward the lowest energy regions, which include the global energy minimum (GEM) structure. The selection procedures tested are the Monte Carlo minimization (MCM) method of Li and Scheraga and the “usage directed” (UD) method of Still's group. LTD is a general method whose parameters can be optimized for any chain system. However,...

Development of a Hybrid Quantum Chemical and Molecular Mechanics Method with Application to Solvent Effects on the Electronic Spectra of Uracil and Uracil Derivatives

A method for molecular dynamics (MD), Monte Carlo (MC), and energy minimization simulation utilizing a Hamiltonian that is divided into two parts is described. One part is treated with a quantum mechanical Hamiltonian, typically a small part of the simulated system that comprises the chromophore. The other part is treated with a classical mechanical Hamiltonian. This partitioning of the system allows us to simulate, for example, not only electronic spectroscopy but also chemical reactions where a bond is broken or to explore the excited state potential energy surface. The particular choice of the quantum mechanical Hamiltonian, the intermediate neglecting of differential overlap (INDO) model Hamiltonian, also offers the possibility of simulating systems that contain a transition metal, which only rarely have been accessible with traditional MD and MC methods. Test calculations on small systems are presented together with an investigation of the photophysics of uracil and 1,3-dimethyluracil.

A new method for modeling large-scale rearrangements of protein domains.

A method for modeling large-scale rearrangements of protein domains connected by a single- or a double-stranded linker is proposed. Multidomain proteins may undergo substantial domain displacements, while their intradomain structure remains essentially unchanged. The method allows automatic identification of an interdomain linker and builds an all-atom model of a protein structure in internal coordinates. Torsion angles belonging to the interdomain linkers and side chains potentially able to form domain interfaces are set free while remaining torsions, bond lengths, and bond angles are fixed. Large-scale sampling of the reduced torsions conformational subspace is effected with the "biased probability Monte Carlo-minimization" method [Abagyan, R.A., Totrov, M.M. (1994): J. Mol. Biol. 235, 983-1002]. Solvation and side-chain entropic contributions are added to the energy function. A special procedure has been developed to generate concerted deformations of a double-stranded interdomain linker in such a way that the polypeptide chain continuity is preserved. The method was tested on Bence-Jones protein with a single-stranded linker and lysine/arginine/ornithine-binding (LAO) protein with a double-stranded linker. For each protein, structurally diverse low-energy conformations with ideal covalent geometry were generated, and an overlap between two sets of conformations generated starting from the crystallographically determined "closed" and "open" forms was found. One of the low-energy conformations generated in a run starting from the LAO "closed" form was only 2.2 A away from the structure of the "open" form. The method can be useful in predicting the scope of possible domain rearrangements of a multidomain protein.

Efficiency of monte carlo minimization procedures and their use in analysis of NMR data obtained from flexible peptides

The Monte Carlo minimization (MCM) method of Li and Scheraga is an efficient tool for generating low energy minimized structures of peptides, in particular the global energy minimum (GEM). In a recent article we proposed an enhancement to MCM, called the free energy Monte Carlo minimization (FMCM) procedure. With FMCM the conformational search is carried out with respect to the harmonic free energy, which approximates the free energy of the potential energy wells around the energy minimized structures (these wells are called localized microstates). In this work we apply both methods to the pentapeptide Leu-enkephalin described by the potential energy function ECEPP, and study their efficiency in identifying the GEM structure as well as the global harmonic free energy (GFM) structure. We also investigate the efficiency of these methods to generate localized microstates, which pertain to different energy and harmonic free energy intervals above the GEM and GFM, respectively. Such microstates constitute an important ingredient of our statistical mechanical methodology for analyzing nuclear magnetic resonance data of flexible peptides. Aspects of this methodology related to the stability properties of the localized microstates are examined. © 1997 by John Wiley & Sons, Inc.

A convenient method for determining cyclic peptide conformation from 1D 1H-NMR information.

A rapid and convenient method for determining the backbone conformation of cyclic peptides results from the combination of 1D 1H NMR information and molecular modeling. phi Angle torsional constraints calculated from 3JHN.H alpha coupling constants are used to determine the position of multiple-welled potential energy penalty functions that are imposed on the force field used in the structure refinement (Amber* with GB/SA solvation model). Monte Carlo searches and minimizations lead to a collection of structures that are clustered by backbone similarity and then filtered according to hydrogen-bonding constraints determined by the chemical shift temperature dependencies of the amide protons. This approach was applied to five cyclic peptides whose structures had been determined previously using more extensive 2D NMR techniques, and the importance of the torsional, H-bonding, and solvation restraints were assessed. For the four peptides that adopt a predominant conformation, this method reproduced the reported structures closely; lack of convergence for the fifth structure reflected the multiple backbone conformations that this macrocycle adopts.

The binding mode of an E-64 analog to the active site of cathepsin B.

Two binding modes of the isobutyl-NH-Eps-Leu-Pro inhibitor to cathepsin B have been proposed. Molecular docking using an empirical force field was carried out to distinguish between the two modes. The search began with manual docking, followed by random perturbations of the docking conformation and cycles of Monte Carlo minimization. Finally, molecular dynamics was carried out for the most favorable docking conformations. The present calculations predict that the isobutyl-NH-Eps-Leu-Pro inhibitor preferentially binds to the S' rather than the S subsites of cathepsin B. The S' binding mode prediction is supported by the X-ray crystal structure of cathepsin B bound to a closely related ethyl-O-Eps-Ile-Pro inhibitor, which was found to bind in the S'subsite with the C-terminal epoxy ring carbon making a covalent bond to the sulfur atom of Cys29. This agreement, in turn, validates our docking strategy. Furthermore, the calculations provide evidence that the dominant contribution to the total stabilization energy of the enzyme-inhibitor complex stems from the strong electrostatic interaction between the negatively charged C-terminal carboxylate group of the ligand and the positively charged imidazolium rings of His110 and His111. The latter are stabilized and held in an optimal orientation for interactions with the C-terminal end of the ligand through a salt bridge between the side chains of His110 and Asp22. By comparison with the crystal structure, some insight into the specificity of the epoxyldipeptide family towards cathepsin B inhibition has been extracted. Both the characteristics of the enzyme (e.g. subsite size and hydrophobicity) as well as the nature of the inhibitor influence the selectivity of an inhibitor towards an enzyme.

Implementation of the ECEPP algorithm, the Monte Carlo minimization method, and the electrostatically driven Monte Carlo method on the Kendall square research KSR1 computer

AbstractIn this article the adaptation of the Empirical Conformational Energy Program for Peptides (ECEPP/3) and two conformational search methods [viz., the Monte Carlo minimization (MCM) method and the electrostatically driven Monte Carlo (EDMC) method] to the Kendall Square Research KSR1 computer is described. The MCM and EDMC methods were developed to surmount the multiple‐minima problem in protein folding. Parallelization of these codes led to substantial speedups (expressed as the ratio between the mean time per energy evaluation in one processor and the mean time per energy evaluation in a set of processors) over the serial versions of these codes. A comparison of the performance of these algorithms on the KSR1 and on the IBM ES9000 computers is presented. © 1995 by John Wiley & Sons, Inc.

A free energy based Monte Carlo minimization procedure for biomolecules

The Monte Carlo minimization (MCM) method of Li and Scheraga (Proc. Natl. Acad. Sci. U.S.A. 1987, 84, 6611) is a highly efficient procedure for generating low-energy structures of macromolecules and therefore has been used extensively. We propose to enhance this method by carrying out the conformational search with respect to the harmonic free energy rather than the energy. This free energy based MCM (FMCM) procedure has been applied initially to the pentapeptide met-enkephalin, modeled by the potential energy function ECEPP. The results demonstrate that FMCM is effective with a negligible extra computational cost and therefore has the potential to become an important computational tool. Thus, within a range of 1 kcal/mol above the global harmonic free energy minimum, structures were generated with energy of up to 8.3 kcal/mol above the global energy minimum (GEM). Also, three conformations were identified with lower free energy than that of the GEM structure. Some of these structures are very unlikely to be detected by the usual MCM method. 27 refs., 1 tab.

ICM—A new method for protein modeling and design: Applications to docking and structure prediction from the distorted native conformation

AbstractAn efficient methodology, further referred to as ICM, for versatile modeling operations and global energy optimization on arbitrarily fixed multimolecular systems is described. It is aimed at protein structure prediction, homology modeling, molecular docking, nuclear magnetic resonance (NMR) structure determination, and protein design. The method uses and further develops a previously introduced approach to model biomolecular structures in which bond lengths, bond angles, and torsion angles are considered as independent variables, any subset of them being fixed. Here we simplify and generalize the basic description of the system, introduce the variable dihedral phase angle, and allow arbitrary connections of the molecules and conventional definition of the torsion angles. Algorithms for calculation of energy derivatives with respect to internal variables in the topological tree of the system and for rapid evaluation of accessible surface are presented. Multidimensional variable restraints are proposed to represent the statistical information about the torsion angle distributions in proteins. To incorporate complex energy terms as solvation energy and electrostatics into a structure prediction procedure, a “double‐energy” Monte Carlo minimization procedure in which these terms are omitted during the minimization stage of the random step and included for the comparison with the previous conformation in a Markov chain is proposed and justified. The ICM method is applied successfully to a molecular docking problem. The procedure finds the correct parallel arrangement of two rigid helixes from a leucine zipper domain as the lowest‐energy conformation (0.5 Å root mean square, rms, deviation from the native structure) starting from completely random configuration. Structures with antiparallel helixes or helixes staggered by one helix turn had energies higher by about 7 or 9 kcal/mol, respectively. Soft docking was also attempted. A docking procedure allowing side‐chain flexibility also converged to the parallel configuration starting from the helixes optimized individually. To justdy an internal coordinate approach to the structure prediction as opposed to a Cartesian one, energy hypersurfaces around the native structure of the squash seeds trypsin inhibitor were studied. Torsion angle minimization from the optimal conformation randomly distorted up to the rms deviation of 2.2 Å or angular rms deviation of l0° restored the native conformation in most cases. In contrast, Cartesian coordinate minimization did not reach the minimum from deviations as small as 0.3 Å or 2°. We conclude that the most promising detailed approach to the protein‐folding problem would consist of some coarse global sampling strategy combined with the local energy minimization in the torsion coordinate space. © 1994 by John Wiley & Sons, Inc.

Detailed ab initio prediction of lysozyme–antibody complex with 1.6 Å accuracy

The fundamental event in biological assembly is association of two biological macromolecules. Here we present a successful, accurate ab initio prediction of the binding of uncomplexed lysozyme to the HyHel5 antibody. The prediction combines pseudo Brownian Monte Carlo minimization with a biased-probability global side-chain placement procedure. It was effected in an all-atom representation, with ECEPP/2 potentials complemented with the surface energy, side-chain entropy and electrostatic polarization free energy. The near-native solution found was surprisingly close to the crystallographic structure (root-mean-square deviation of 1.57 A for all backbone atoms of lysozyme) and had a considerably lower energy (by 20 kcal mol-1) than any other solution.

Atomistic simulation of void nucleation in aluminium lines

Abstract The formation of voids in aluminium interconnects in integrated circuits was studied atomistically with emphasis on nucleation. Input computational cells included different levels of concentrations of vacancies and stresses together with the typically observed structural defects (grain boundaries and dislocations) in metal lines. The input cells were relaxed using Monte Carlo (atom transpose) and static energy minimization (strain relaxation) techniques concurrently. Besides agglomeration of vacancies in unstrained crystals, the formation of vacancies in crystals under tension was found to be a main contributor to void nucleation. Climb and glide of dislocations were observed atomistically during the formation of voids. Compressive stresses in the metal lines, on the other hand, prevented the formation of voids.

Monte Carlo docking of oligopeptides to proteins

A new two-step procedure has been developed for the docking of flexible oligopeptide chains of unknown conformation to static proteins of known structure. In the first step positions and conformations are sampled and the association energy minimized starting from an approximate preselected docking position. The resulting conformations are further optimized in the second step by a Metropolis Monte Carlo minimization, which optimizes each of these structures. The method has been tested on the HIV-1 aspartic proteinase complex with an inhibitor, whose crystallographic structure is known at 2.3 A resolution. Furthermore, the application of this method to the docking of the hendecapeptide 58-68 of the influenza A virus matrix protein to the HLA-A2 molecule produced results which are in agreement with experimental observations in identifying side chains critical for T cell recognition and residues responsible of MHC protein binding.

Structure and free energy of complex thermodynamic systems

Biological systems are intrinsically complex, involving many degrees of freedom, heterogeneity, and strong interactions among components. For the simplest of biological substances, e.g., biomolecules, which obey the laws of thermodynamics, we may try to investigate them by means of a statistical mechanical approach. Even for these simplest many-body systems, assuming all microscopic interactions are completely known, current physical and chemical methods of characterizing the overall structure and free energy face the fundamental challenge of an exponential amount of computations as the number of degrees of freedom of these systems increases. As a response to this problem, two general procedures have been developed to compute the structure (Monte Carlo-minimization method) and free energy (Monte Carlo recursion method) of a complex thermodynamic system. The Monte Carlo-minimization procedure has been applied to determine the structure of a pentapeptide Met-enkephalin, leading consistently to a stable β-bend structure, starting from random initial conformations. The Monte Carlo recursion method has been applied to a Lennard-Jones fluid, with results in agreement with previously published values of the free energy obtained from other procedures.

Monte Carlo-minimization approach to the multiple-minima problem in protein folding.

A Monte Carlo-minimization method has been developed to overcome the multiple-minima problem. The Metropolis Monte Carlo sampling, assisted by energy minimization, surmounts intervening barriers in moving through successive discrete local minima in the multidimensional energy surface. The method has located the lowest-energy minimum thus far reported for the brain pentapeptide [Met5]enkephalin in the absence of water. Presumably it is the global minimum-energy structure. This supports the concept that protein folding may be a Markov process. In the presence of water, the molecules appear to exist as an ensemble of different conformations.

Application of empirical ionic models to SiO2 liquid: Potential model approximations and integration of SiO2 polymorph data

Structural and thermodynamic properties of crystalline SiO2 and SiO2 liquid have been examined with Monte Carlo (MC), molecular dynamics (MD), and energy minimization (EM) calculations using several ionic potential models obtained from the literature. The MC and MD methods calculate the same structural and thermodynamic properties for liquids when the same potential model is used. The Ewald (1921) method of calculating coulomb interactions reproduced most successfully the structure of liquid silica. Approximating the coulomb interaction by eliminating the inverse lattice sum results in predicted bond distances that are too short and an average 〈Si-O-Si〉 angle of approximately 180°. Introduction of a cut-off in the potential energy function produces irregular tetrahedra and inconsistencies in predicted Si-O coordination in silica liquid. The system internal energies show that liquid structures derived from random starting configurations can be metastable relative to structures calculated from crystalline starting configurations. The static lattice properties of the polymorphs alpha-quartz, coesite, and stishovite were used to evaluate further the accuracy of different sets of repulsive parameters for the full Ewald ionic model. Most of the models studied reproduced poorly the measured structures and elastic constants of the polymorphs. The major weakness of the ionic model is the unreasonably large Si-O bond strength (120 × 10−12 ergs/bond) when formal ionic charges are used. Fractional charge models with a small Si-O bond strength (30 × 10−12 ergs/bond) improve the agreement with experimental data. However, further improvement of the ionic model should include reducing the Si-O bond strength to values in better agreement with published estimates (7 × 10− 12 to 13 × 10−12 ergs/bond). By using additional information to constrain the parameterization of the ionic model, such as estimated bond strengths and static properties of the silica polymorphs, a model more representative of the interparticle interactions may be obtained.

Thermodynamic Models of Liquid Silicates

Abstract The development of structure-based thermodynamic models of binary silicates, which has occurred since the publications of Toop and Samis (1962) and Masson (1965), is reviewed. The various approaches used in the construction of these models include consideration of generalized polymerization-depolymerization equilibria involving the three forms of oxygen occurring in silicate melts and involving specificequilibria among discrete chain silicate ions and free oxygen ions, the application of the principles of polymer theory to calculation of the most probable distribution of branching chain silicate ions, calculation of the mixing properties by statistical mechanics. Monte Carlo minimization of free energy, and statistical approaches which consider the distribution of silicon and oxygen in quasi-lattices.The various models are examined and compared. Direct comparison is facilitated by the fact that, in the construction of each of the models, the free energy change for the depolymerization reaction O°...